Digital Signal Processing Reference
In-Depth Information
That is to say in decibels:
()
⎛
BT
⎞
=
−
3dB
()
W
W
03
−
[2.80]
⎜
⎟
dB
dB
2
⎝
⎠
According to the scale change theorem, a time dependent expansion of a factor
a
> 0 can be translated by a frequential contraction of a factor
1
a
. Thus, we can
standardize this bandwidth in the following way:
()
B
=
T B
T
[2.81]
−
3dB
−
3dB
This reduced bandwidth is independent of the amplitude and the length
T
of the
window support, and it depends only on its shape.
Another measurement of the width of the main lobe is the equivalent noise
bandwidth
B
en
(
T
) of the bandwidth
.
It is the bandwidth of the window whose
spectral energy density is rectangular, having the same integral and same maximum
value
0,
ˆ
w
(0) as the density of the window considered (the spectral energy density
of a real positive function is always maximum for the zero frequency). We
immediately obtain the expression of this bandwidth, in the frequential and time
dependent fields (by using Parseval's theorem):
+∞
2
∫
()
ˆ
wf
f
0,
T
−∞
()
BT
=
en
0,
()
ˆ
w
0
T
T
∫
0,
()
wt t
T
0
=
2
⎛
T
⎞
∫
()
wt t
⎜
⎟
0,
T
⎝
0
⎠
Here too we can define the reduced bandwidth independent of
T
:
()
B
BT
[2.82]
en
en
The second characteristic of a window is the amplitude of the secondary lobes in
relation to the amplitude of the main lobe. A natural measurement is the amplitude
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